2. Ways to deal with Big Data
• Big Analytics - use distributed database systems
(hadoop) and parallel programming
(MapReduce)
• Sampling - use the representative sample
estimate the population
• Sampling in Hadoop
• Hadoop isn’t the king of interactive analysis
• Sampling is a good way to grab a set of data then
play with it locally (R or Excel)
• Pig has a handy SAMPLE keyword
3. Elements of a Sample
• Sample - a subset of individuals within a statistical population to
estimate characteristics of the whole population.
• Target Population - collection of observations we want to study
• Sampled Population - all possible observation units that might
have been sampled
• Sampling Frame - list of all sampling units (student roster, list of
phone number)
• Sampling Unit - unit we actually sample (e.g. household)
• Observational Unit - element to be measured (e.g. individual
people in the household)
4. Sampling Techniques (1)
• Probability Sampling
• Every unit in the population has a chance (greater than zero) of
being selected in the sample, and this probability can be
accurately determined.
• Not every observational unit has to have the same probability of
selection but every observational unit’s probability is known.
• Nonprobability Sampling
• Some elements of the population have no chance of selection
(these are sometimes referred to as 'out of coverage'), or where
the probability of selection can't be accurately determined.
• Because the selection of elements is nonrandom, nonprobability
sampling does not allow the estimation of sampling errors.
6. Simple Random Sampling - SRS
• Definition: for a size n simple random sample, every possible
subset of n units in the population has the same chance of
being in the sample
• Requirement: One unique identifier is needed for
implementation
• Advantage: easy to understand and implement
• Disadvantage: biggest variance, least accuracy
7. Systematic Sampling
• Definition: Systematic sampling relies on arranging the study
population according to some ordering scheme and then
selecting elements at regular intervals through that ordered
list. Systematic sampling involves a random start and then
proceeds with the selection of every kth (k=population
size/sample size) element from then onwards.
• Requirement: Ordering scheme for population
• Advantage: easy to implement, very efficient
• Disadvantage: vulnerable to periodicities
8. Stratified sampling (1)
• Definition: Where the population embraces a number of
distinct categories, the frame can be organized by these
categories into separate "strata." Each stratum is then
sampled as an independent sub-population, out of which
individual elements can be randomly selected.
• Requirement: population can be divided into distinct,
independent strata, provided that strata are selected based
upon relevance to the criterion in question
• Variability within strata are minimized
• Variability between strata are maximized
• The variables upon which the population is stratified are
strongly correlated with the desired dependent variable.
9. Stratified sampling (2)
• Advantage:
• Inferences can be done about specific subgroup
• Very likely more efficient statistical estimates
• will never result in less efficiency than SRS, provided that each
stratum is proportional to the group's size in the population.
• Data maybe more readily for individual pre-existing strata within
a population than for the overall population
• Because strata are independent, different approaches for
subgroups
• Disadvantage:
• Complexity in implementation and estiamtion
• Multiple criteria can be tricky
• Specified minimum sample size per group
10. Cluster Sampling (1)
• Definition: where the entire population is divided into
groups, or clusters, and a random sample of these clusters are
selected. All observations in the selected clusters are included
in the sample.
• Requirement: does not require complete list of every unit in
the population, only requires sampling frame on cluster-level
• Variability within cluster are maximized
• Variability between cluster are minimized
• The variables upon which the population is divided into
clusters are not strongly correlated with the desired
dependent variable.
11. Cluster Sampling (2)
• Advantages:
• Easy to implement
• Cost-effective
• Disadvantages:
• Complexity in estimation
• May not reflect the diversity of clusters
• Provide less information per observation than SRS
• Redundant information from the others in the cluster
• Standard errors may be higher than other sampling designs
12. Probability Proportional to size
sampling - PPS
• Definition: Where the selection probability for each element is
set to be proportional to its size measure.
• Every technique before was equal probability of selection (EPS)
• Requirement: auxiliary variable / size measure, correlated to
the variable of interest
• Advantage:
• May improve accuracy for a given sample size by concentrating
sample on large elements that have the greatest impact on
estimation
• For business and auditing, monetary unit sampling (MUS)
• Disadvantage:
• Complexity for implementation and estimation
• Different portions of the population may be over or under
represented due to the probability variation in selection
13. Representativeness of the sample
• Match between target population and
sampled population
• Method of drawing sample
14. Two kinds of Errors
• Non-sampling error - can be reduced by careful design of the survey
• Selection bias - part of target population is not in sampled population
(target population may not have a natural frame, the mode of data
collection may restrict frame)
• Coverage Error - the extent to which the Sampling Frame does not cover
the Target population
• Measurement bias - measuring instrument has tendency to differ
from true value in one direction
• Measurement error (Errors of Observation)
•
•
•
•
Deviations of measurement
Inaccurate measurement
Item nonresponse (didn’t understand, didn’t see, or refused question)
Unit nonresponse (not home, not approached by interviewer, refuse call)
• Sampling error - results from taking a sample instead of whole
population, can be quantified by statistics, reduced by increasing
sample size
15. Sample Size Calculation
• In order to know what our sample size needs to be, we must
decide in advance the maximum estimation error we are
willing to tolerate.
• Determine the nature of estimation – proportion or mean
• The confidence level of your estimation – significant level
16. Proportion (1)
• Proportion: p^ = X/n
• where X is the number of 'positive' observations, n is sample size
• When the observations are independent, the estimator has a
binomial distribution, variance = np(1-p)
• The maximum variance of this distribution is 0.25*n, when
p=0.5
• For sufficiently large n, the distribution of p^ will be closely
approximated by a normal distribution. around 95% of this
distribution's probability lies within 2 standard deviations of
the mean.
• will form a 95% confidence interval for the true proportion.
17. Proportion (2)
• If this interval needs to be no more than W units wide, the
equation
• can be solved for n, yielding n = 4/W2 = 1/B2 where B is the
error bound on the estimate
• i.e., the estimate is usually given as within ± B. So,
• for B = 10% one requires n = 100,
• for B = 5% one needs n = 400,
• for B = 3% the requirement approximates to n = 1000,
• while for B = 1% a sample size of n = 10000 is required.
18. Mean (1)
• A proportion is a special case of a mean. When estimating the
population mean using an independent and identically
distributed (iid) sample of size n, where each data value has
variance σ2, the standard error of the sample mean is:
• This expression describes quantitatively how the estimate
becomes more precise as the sample size increases. Using the
central limit theorem to justify approximating the sample
mean with a normal distribution yields an approximate 95%
confidence interval of the form
19. Mean (2)
• If we wish to have a confidence interval that is W units in
width, we would solve
• for n, yielding the sample size n = 16σ2/W2.
• i.e., if we are interested in estimating the amount by which a
drug lowers a subject's blood pressure with a confidence
interval that is 6 units wide, and we know that the standard
deviation of blood pressure in the population is 15, then the
required sample size is 100
20. Stratified Sample Size (1)
• The sample can often be split up into sub-samples. Typically, if
there are k such sub-samples (from k different strata) then
each of them will have a sample size ni, i = 1, 2, ..., k. These ni
must conform to the rule that n1 + n2 + ... + nk = n (i.e. that
the total sample size is given by the sum of the sub-sample
sizes). Selecting these ni optimally can be done in various
ways, using (for example) Neyman's optimal allocation.
• There are many reasons to use stratified sampling:[7] to
decrease variances of sample estimates, to use partly nonrandom methods, or to study strata individually. A
useful, partly non-random method would be to sample
individuals where easily accessible, but, where not, sample
clusters to save travel costs.
